What a representation of a group is please?

[robousy]

Can please explain to me what a representation of a group is please?

Hopefully something more illuminating than what I might find on wikipedia or eric weissteins mathworld.

Also, perhaps illustrated with an example.

 

[Hurkyl]

Consider the group of rotations about a given point in the plane. One representation of this group is SO(2), the collection of 2x2 orthogonal matrices with determinant 1.

SO(2), incidentally, is also a representation of the group of translations along a given line. (e.g. take the mapping that a translation of t units is represented as the matrix that represents a rotation of t radians about the origin) This one isn't faithful.

 

[robousy]

aaaaah.

So, when I talk about SO(2), SU(2), SU(3), SU(5), SU(10) etc then they are representations of a BIGGER group.

SO(2) is not the group - its part of the group. I think I am starting to see now.

So, the standard rotation matrix for SO(2) with the sins and the cosine terms is a representation.

 

[Hurkyl]

Well, the other group isn't always bigger -- for example, SO(3) is a perfectly good representation of the group SO(3). (But, of course, SO(3) has representations in higher-dimensional matrix groups too)

 

[robousy]

Ok, I think I have another way of stating it:

SO(2) is a REPRESENTATION of the GROUP of 2 by 2 Real matrices.

Is this accurate?

 

[Hurkyl]

SO(2) is a (particular) group of 2x2 real matrices.

SO(2) is a representation of the group of planar rotations about a specified point.

There are representations of SO(2) that are in a higher-dimensional matrix group.

 

[robousy]

So you can have representations of SO(2) that are not 2 by 2 matrices?

 

[Hurkyl]

Yes you can.

For example, there is an important representation of SU(2) as a group of real matrices. (Specifically, SO(3) is a representation of SU(2))

 

[robousy]

SO(2) is a (particular) group of 2x2 real matrices.

SO(2) is a representation of the group of planar rotations about a specified point.

There are representations of SO(2) that are in a higher-dimensional matrix group.

Thanks for all your patience. I'm taking a GUT course at the moment in QFT but there is so much group theory that I am not 'getting' a lot of it.

Now, above you say that SO(2) is a representation of the group ...

I hope this is not pedantic but is it a or the (one and only) representation of the group of planar rotations about a specified point.

I am probably displaying my ignorance here but how many other ways are there of representing a rotation?

 

[Hurkyl]

Well, there are trivial representations -- all rotations could be mapped to, say, the zero 7x7 matrix.

The rotations of the plane could be represented as simultaneous rotations along two orthogonal planes in 4-dimensional space.

Planar rotations could be represented as operations on the vector space of all functions on the plane -- a rotation R is represented by the operator T defined by Tf := fR. (that is, (Tf)(x) := f(Rx))

(I can't think of any interesting ones -- someone said somewhere else that SO(2) doesn't have any interesting representations)

 

[robousy]

Ok, how is this summary:

SO(2) is a representation of the group of 2D rotations about a point,
SO(3) is a representaion of the group of 3D rotations about a point.
SU(2) is a representation of the group of rotations in a 4 (?) D complex space

etc...

Now, how does a representation differ from a fundamental representation?

 

[Hurkyl]

I don't think any group of rotations can be represented by SU(2). (don't quote me on that, though)

SU(2) is a subgroup of the invertible 2x2 complex matrices, so such things act on 2-dimensional complex vector spaces (which are 4-dimensional real vector spaces).


Some representations can be broken into parts. For example, I can represent the multiplicative group of positive real numbers as the group of all 2x2 real diagonal matrices with two equal positive diagonal entries. It's clear that we can decompose R² into two subspaces (projection onto first coordinate and projection onto second coordinate) and we don't lose anything about the group representation. We say that this representation is reducible.

(Incidentally, the two pieces that we split this one into would be irreducible)

I'm not sure if fundamental is just another word for irreducible, or if it's something else.

 

[robousy]

For example:

$$\left(\begin{array}{cc}3 & 0\\0 & 3\end{array}\right) \times \left(\begin{array}{cc}5 & 0\\0 & 5\end{array} \right) \: = \: \left(\begin{array}{cc}15 & 0\\0 & 15\end{array}\right)$$

and the irreducible parts of these would be:

$$\left(\begin{array}{cc}3\\0\end{array}\right) , \left(\begin{array}{cc}0\\3\end{array}\right) , \left(\begin{array}{cc}5\\0\end{array}\right) , \left(\begin{array}{cc}0\\5\end{array}\right)$$


Is this what we mean by the irreducible rep?

 

[matt grime]

Try looking at my web page for some group theory stuff.

Personally, I don't like the ideas that this thread has gotten across.

SO(2) *is* a group, it is vacuously a representation of itself.

A representation of a group G is a homomoprhism from G into the set of nxn matrices over some field. Some people would describe that as a linaerization of the group.

Every group has a trivial (technical term) representation (herein called rep) that sends any g in G to 1, thinking of the field (assume we mean complex numbers from now on) as a set of 1x1 matrices.

Any permutation group on n elements, S_n has a natural representation as a set of nxn matrices. Pick a basis

e_1,e_2,..,e_n of C^n and let a permutation permute the subscripts.

This vector space has a natural 1-d subspace that is left invariant by all of the elements of G, namely the line spanned by

e_1+e_2+..+e_n

Call this line L, then C^n/L is also a representation of S_n.

The cylcic group C_n has n 1-dimensional complex representations. Send the generator of C_n to some n'th root of unity.

The group given by

G= <s,t : s^2=t^n=e sts=t^{-1} >

has a natural 2-d representation since I've just given you a *presentation* of the dihedral group. We can easily make more representations from this.

In particular if p(g) is some homomorphism from G to End(V) the nxn matrices (V a vector space of dim n) then det(p(g)) is a representation. Thus send the rotation matirices to +1 and the refelction matrices to -1 is a rep of D_{2n} too.

I am very unsure what your last post was about at all.

A representation is irreducible if it has no subspaces that are also representations by restriction. Or it has no subspaces that are invariant by the action of G.

What you wrote out was a matrix multiplication then some of its components/columns.

 

[Hurkyl]

Yay! Thanks for responding matt, I really don't know much about this topic, and was hoping someone would chime in eventually!

 

[matt grime]

probably won't have time to respond much any more to posts, but i wanted to get across some important ideas.

SO(2) is the group of rotations, it is trivially a representation of itself by sending the matrix g to the matrix g in End(R^2). i would call that a degenerate case. better is to find a vector space on which something acts that is slightly surprising (I think that actually all lie groups turn out to have "unsurprising" reps since they cvan be formed from the 2-d rep via some tensor algebra, thoguh i am extrapolating this from only partial knowledge of weight spaces for finite chevally groups)

There are other representations of it

SU(2) is a group of complex matrices over C. All its (finite dimensional) reps over C are easy to describe: they are the set of n+1 degree homogeneuous polys in 2 dimensions and can easily be found in many discussions on the web since they are the best group for explaining the representation theory of complex lie groups and are of interest to many people.

 

[robousy]

Hey Matt.

Thanks for your long response earlier.

I put off reading it as it intimidated me and know I have read it I am sorry to say I don't understand any of it.

I need (and want) to however to help with my graduate field theory classes.

What sort of maths is this? I think its abtract algebra which I have no training in.

I will learn this.

Thanks again and sorry it fell on dumb ears.

 

[matt grime]

Let G be a group. Let GL(C,n) be the group of invertible nxn matrices with entries in C. You know what all these things mean. Let p be a homomorphism (you konw what a homomorphism is) from G to GL(n,C). This is a representation of G. It is a way of making G act by matrices on some finite dimensional vector space. (We can do it for infinite dimensional spaces too but I'd rather not explain those. not because they are harder but because they come in two flavours and i wouldn't want to talk about the wrong one. I'm sure we'll see which is more useful to you later; probably you'll care about unitary representations if at all).So, to recap, to understand representation theory you need to understand

G a group
GL(n,C) the invertible nxn matrices (also a group)
p:G-->GL(n,C) a group homomorphism.

the map p is the representation (some times we refer to C^n as the representation and leave the p implicit)

which of those is confusing?

SO(2) is a group. There is a natural and obvious representation where we identify an element of SO(2) with itself inside GL(2,C)

We can also create a map p:SO(2)-->GL(4,C) by making the matix act on the pair of vecotrs (u,v) (Ie think of u and v as vectors in C^2) by p(g)(u,v)=(gu,gv)
the tiitle "representation theory" is a very big one. It encompasses a great many things in (predominantly) algebra.

A representation of some object (be it a group or whatever) is a way of realizing this thing as a set of matrices (though we possibly lose information, like in the trivial representation of any group)

as long as you come away with a simple idea that a rep of a group is a way of interpreting the group as matrices (ie making it a concrete set of transformations of some vector space) then you're more than halfway to understanding what a representation is.

The best answer is to buy this book:



or if that link is broken representations and characters of groups by James and Loebeck. that is the hard cover and is stupidly priced, try to find the soft cover or get your library to buy it.

i will try and answer more questions over the weekend if tyou can come up with a list of queries that are bothering yuo

 

[robousy]

Hi Matt and thanks for taking the time out to explain this - Its probably really trivial for you so thanks for your patience.

Ok, what's confusing me is that I just find it easier to take on a concept or idea if I see an example involving actual matrices. Sadly I'm not at that level yet where I can just take things in without seeing it laid out.

I'll try and give you a silly example and you'll hopefully see where my thinking is going wrong.

let G be the SO(2) rotation grp.

let p (the homomorphism) be the identity matrix time a real constant. (so the diagonal can be 1,1 or 2,2 or 3,3 etc and off diag is zero.

p:G-->GL(n,C)

I know this is a silly example but my questions here really are:

1) is p a representation in this example (even through quite trivial)?
2) p can clearly take on an infinity of values in this case so are there an infinite number of reps for a given group?

Thanks again for your patience.

 

[George Jones]

I think you need to slow down - you can't learn everything all at once.

Progress in understanding mathematics relies on definitions in at least 2 ways: 1) having some idea of what a particular defintion is trying to say; 2) being able to work with a particular definition.

I think definitions are good starting points here. After the definitions are cleared up, we can move on to using them.

What is a group?

What does p:G-->GL(n,C) mean?

What is a homomorphism?

What is a representation of a group?

Regards,
George

 

[matt grime]

From your "example" it is clear to me tha\t you don't know what a homomorphism is. Is that a correct deduction? Start by thinkiong of finite groups as they are easier.

If you insist on thinking of SO(2,R) then why do you think sending any element to diag(2x2) can be a homomorphism (see lead in question)? It can't: it doesn't even send the identity to the identity.

The map sending any matrix in SO(2) to the identity in GL(n,C) for any n is a representation. Not a very interesting one, admittedly.

Yes, there are an infinite number of (essentially different) representations, and some representation may appear in different forms even though they are the same. NB we declare to representations to be the same if they only differ by a change of basis (eg if we talk the natural rep of SO(2) including it into GL(2,C), all that p(g)=g then all the represetnations q(g) = XyX^{-1} are "the same" because we have just changed basis.

 

[robousy]

I think you need to slow down - you can't learn everything all at once.
Progress in understanding mathematics relies on definitions in at least 2 ways: 1) having some idea of what a particular defintion is trying to say; 2) being able to work with a particular definition.
I think definitions are good starting points here. After the definitions are cleared up, we can move on to using them.
What is a group?
What does p:G-->GL(n,C) mean?
What is a homomorphism?
What is a representation of a group?
Regards,
George

Ok, good approach:

GROUP

A group is a set that satisfy associativity, that have an inverse element, identity element and closure. I understand these.


p:G-->GL(n,C)


Well, I ripped this from Matt Grimes thread and used it in a way I thought it worked, that being p: is some mapping instruction that maps whatever was in G into GL(n,c) - if I am in error then I apologise.

Homomorphism

I think its an instruction to map one matrix into another.
In my example I thought that a map might be to multiply by some constant times the identity. (boring I know but just to get the idea of what constituted a representation).

Representation

The 6 million dollar question! :)
This is what I am trying to come to an appreciation of.

 

[robousy]

From your "example" it is clear to me tha\t you don't know what a homomorphism is. Is that a correct deduction?

$$True$$

If you insist on thinking of SO(2,R) then why do you think sending any element to diag(2x2) can be a homomorphism (see lead in question)? It can't: it doesn't even send the identity to the identity.

My java tex was messing up at the time of writing. Its ok now.
What I meant was
$$\left(\begin{array}{cc}a & b\\c & d\end{array}\right) \left(\begin{array}{cc}2 & 0 \\0 & 2\end{array}\right) = \left(\begin{array}{cc}2a & 2b\\2c & 2d\end{array}\right)$$
Ok - clearly I don't claim to be an expert (or anywhere near it) but I am trying to understand the idea of a homomorphism - expressing to you what I 'think' is going on, and waiting to be corrected.
I 'think' here the matrix containing a,b,c,d is 'mapped' via the matrix (2,0,0,2) into the 'representation' 2a,2b,2c,2d.
I know this is all very trivial but I need to get my definitions correct first.

Yes, there are an infinite number of (essentially different) representations, and some representation may appear in different forms even though they are the same.

Thanks for clearing that up and thanks again for your patience.

 

[matt grime]

No, no and thrice no.

Until you learn what a group and a homomorphism is, say by reading any basic textbook about algebra, there genuinely is no point in you trying to think about representation theory since a representation IS a homomorphism from G a group to a group of matrices. If you don't know what one of those is you *can't* understand representation theory.

 

[Ruslan_Sharipov]

The best answer is to buy this book ... or if that link is broken representations and characters of groups by James and Loebeck. That is the hard cover and is stupidly priced, try to find the soft cover or get your library to buy it.

There is a free option. Visit my site http://www.freetextbooks.boom.ru and download my book "Representations of finite groups" for free from there either in English or in Russian. The same book is also available here:

http://arXiv.org/abs/math/0612104

With Happy New Year greetings on behalf of Santa,
yours sincerely Ruslan Sharipov +7(917)476-93-48

Dec. 29, 2006.

 

[Chris Hillman]

representations and characters of groups by James and Loebeck.

You mean Gordon James and Martin Liebeck, Representations and Characters of Groups, Cambridge University Press, 1993. An excellent and very readable textbook which is designed for self-study, so it should be ideal for the OP (after say Herstein, Abstract Algebra, MacMillan, 1986, which is short and sweet and also very readable).

IMHO, CUP actually offers very reasonable prices, compared to say Kluwer.